Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in five and more dimensions

TL;DR

This paper proves small data global well-posedness and scattering for the Klein-Gordon-Zakharov system in five or more dimensions at the critical regularity using $U^2, V^2$ spaces.

Contribution

It establishes the critical regularity well-posedness and scattering results for the Klein-Gordon-Zakharov system in high dimensions, extending previous understanding.

Findings

Global well-posedness at critical regularity in $d \\ge 5$

Scattering results for small initial data

Use of $U^2, V^2$ function spaces for analysis

Abstract

We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s_c=d/2-2$. By $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 5$.